<!DOCTYPE html>
<html lang="zh-CN">
<head>
  <meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
<meta name="theme-color" content="#222">
<meta name="generator" content="Hexo 4.2.1">
  <link rel="icon" type="image/png" sizes="32x32" href="/images/32x32.png">
  <link rel="icon" type="image/png" sizes="16x16" href="/images/16x16.png">

<link rel="stylesheet" href="/css/main.css">

<link rel="stylesheet" href="//fonts.googleapis.com/css?family=Lato:300,300italic,400,400italic,700,700italic|Lato', 'Microsoft Yahei Light:300,300italic,400,400italic,700,700italic|Cambria', 'Microsoft Yahei Light:300,300italic,400,400italic,700,700italic|Verdana', Lato, 'Microsoft Yahei Light:300,300italic,400,400italic,700,700italic&display=swap&subset=latin,latin-ext">
<link rel="stylesheet" href="/lib/font-awesome/css/all.min.css">

<script id="hexo-configurations">
    var NexT = window.NexT || {};
    var CONFIG = {"hostname":"yoursite.com","root":"/","scheme":"Mist","version":"7.8.0","exturl":false,"sidebar":{"position":"right","display":"hide","padding":18,"offset":12,"onmobile":false},"copycode":{"enable":false,"show_result":false,"style":null},"back2top":{"enable":true,"sidebar":false,"scrollpercent":false},"bookmark":{"enable":false,"color":"#222","save":"auto"},"fancybox":false,"mediumzoom":false,"lazyload":false,"pangu":false,"comments":{"style":"tabs","active":null,"storage":true,"lazyload":false,"nav":null},"algolia":{"hits":{"per_page":10},"labels":{"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}},"localsearch":{"enable":true,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":false},"motion":{"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}},"path":"search.xml"};
  </script>

  <meta name="description" content="Problem 156 Counting Digits Starting from zero the natural numbers are written down in base 10 like this:0 1 2 3 4 5 6 7 8 9 10 11 12…. Consider the digit d&#x3D;1. After we write down each number n, we w">
<meta property="og:type" content="article">
<meta property="og:title" content="Problem 156">
<meta property="og:url" content="http://yoursite.com/156/index.html">
<meta property="og:site_name" content="Project Euler | 欧拉计划">
<meta property="og:description" content="Problem 156 Counting Digits Starting from zero the natural numbers are written down in base 10 like this:0 1 2 3 4 5 6 7 8 9 10 11 12…. Consider the digit d&#x3D;1. After we write down each number n, we w">
<meta property="og:locale" content="zh_CN">
<meta property="article:published_time" content="2007-05-26T05:00:00.000Z">
<meta property="article:modified_time" content="2015-10-20T11:38:18.000Z">
<meta property="article:author" content="sx349">
<meta name="twitter:card" content="summary">

<link rel="canonical" href="http://yoursite.com/156/">


<script id="page-configurations">
  // https://hexo.io/docs/variables.html
  CONFIG.page = {
    sidebar: "",
    isHome : false,
    isPost : true,
    lang   : 'zh-CN'
  };
</script>

  <title>Problem 156 | Project Euler | 欧拉计划</title>
  






  <noscript>
  <style>
  .use-motion .brand,
  .use-motion .menu-item,
  .sidebar-inner,
  .use-motion .post-block,
  .use-motion .pagination,
  .use-motion .comments,
  .use-motion .post-header,
  .use-motion .post-body,
  .use-motion .collection-header { opacity: initial; }

  .use-motion .site-title,
  .use-motion .site-subtitle {
    opacity: initial;
    top: initial;
  }

  .use-motion .logo-line-before i { left: initial; }
  .use-motion .logo-line-after i { right: initial; }
  </style>
</noscript>

</head>

<body itemscope itemtype="http://schema.org/WebPage">
  <div class="container use-motion">
    <div class="headband"></div>

    <header class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-container">
  <div class="site-nav-toggle">
    <div class="toggle" aria-label="切换导航栏">
      <span class="toggle-line toggle-line-first"></span>
      <span class="toggle-line toggle-line-middle"></span>
      <span class="toggle-line toggle-line-last"></span>
    </div>
  </div>

  <div class="site-meta">

    <a href="/" class="brand" rel="start">
      <span class="logo-line-before"><i></i></span>
      <h1 class="site-title">Project Euler | 欧拉计划</h1>
      <span class="logo-line-after"><i></i></span>
    </a>
  </div>

  <div class="site-nav-right">
    <div class="toggle popup-trigger">
        <i class="fa fa-search fa-fw fa-lg"></i>
    </div>
  </div>
</div>




<nav class="site-nav">
  <ul id="menu" class="main-menu menu">
        <li class="menu-item menu-item-home">

    <a href="/" rel="section"><i class="fa fa-home fa-fw"></i>首页</a>

  </li>
        <li class="menu-item menu-item-problem">

    <a href="/problems" rel="section"><i class="fa fa-tags fa-fw"></i>题目</a>

  </li>
        <li class="menu-item menu-item-about">

    <a href="/about/" rel="section"><i class="fa fa-user fa-fw"></i>关于</a>

  </li>
      <li class="menu-item menu-item-search">
        <a role="button" class="popup-trigger"><i class="fa fa-search fa-fw"></i>搜索
        </a>
      </li>
  </ul>
</nav>



  <div class="search-pop-overlay">
    <div class="popup search-popup">
        <div class="search-header">
  <span class="search-icon">
    <i class="fa fa-search"></i>
  </span>
  <div class="search-input-container">
    <input autocomplete="off" autocapitalize="off"
           placeholder="搜索..." spellcheck="false"
           type="search" class="search-input">
  </div>
  <span class="popup-btn-close">
    <i class="fa fa-times-circle"></i>
  </span>
</div>
<div id="search-result">
  <div id="no-result">
    <i class="fa fa-spinner fa-pulse fa-5x fa-fw"></i>
  </div>
</div>

    </div>
  </div>

</div>
    </header>

    
  <div class="back-to-top">
    <i class="fa fa-arrow-up"></i>
    <span>0%</span>
  </div>


    <main class="main">
      <div class="main-inner">
        <div class="content-wrap">
          

          <div class="content post posts-expand">
            

    
  
  
  <article itemscope itemtype="http://schema.org/Article" class="post-block" lang="zh-CN">
    <link itemprop="mainEntityOfPage" href="http://yoursite.com/156/">

    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="image" content="/images/avatar.png">
      <meta itemprop="name" content="sx349">
      <meta itemprop="description" content="Project Euler | 欧拉计划 中文翻译站">
    </span>

    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="Project Euler | 欧拉计划">
    </span>
      <header class="post-header">
        <h1 class="post-title" itemprop="name headline">
          Problem 156
        </h1>

        <div class="post-meta">
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="far fa-calendar"></i>
              </span>
              <span class="post-meta-item-text">题目发布于</span>

              <time title="发布时间：2007-05-25 22:00:00" itemprop="dateCreated datePublished" datetime="2007-05-25T22:00:00-07:00">2007-05-25</time>
            </span>
              <span class="post-meta-item">
                <span class="post-meta-item-icon">
                  <i class="far fa-calendar-check"></i>
                </span>
                <span class="post-meta-item-text">翻译更新于</span>
                <time title="更新时间：2015-10-20 04:38:18" itemprop="dateModified" datetime="2015-10-20T04:38:18-07:00">2015-10-20</time>
              </span>

          

        </div>
      </header>

    
    
    
    <div class="post-body" itemprop="articleBody">

      
        <hr>
<h1 id="Problem-156"><a href="#Problem-156" class="headerlink" title="Problem 156"></a><a href="https://projecteuler.net/problem=156" target="_blank" rel="noopener">Problem 156</a></h1><hr>
<p><b>Counting Digits</b></p>
<p>Starting from zero the natural numbers are written down in base 10 like this:<br>0 1 2 3 4 5 6 7 8 9 10 11 12….</p>
<p>Consider the digit d=1. After we write down each number n, we will update the number of ones that have occurred and call this number f(n,1). The first values for f(n,1), then, are as follows:</p>
<table>
<thead>
<tr>
<th align="center">n</th>
<th align="center">f(n,1)</th>
</tr>
</thead>
<tbody><tr>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">2</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">7</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">8</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">9</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">10</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">11</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">12</td>
<td align="center">5</td>
</tr>
</tbody></table>
<p>Note that f(n,1) never equals 3.<br>So the first two solutions of the equation f(n,1)=n are n=0 and n=1. The next solution is n=199981.</p>
<p>In the same manner the function f(n,d) gives the total number of digits d that have been written down after the number n has been written.<br>In fact, for every digit d ≠ 0, 0 is the first solution of the equation f(n,d)=n.</p>
<p>Let s(d) be the sum of all the solutions for which f(n,d)=n.<br>You are given that s(1)=22786974071.</p>
<p>Find  ∑ s(d) for 1 ≤ d ≤ 9.</p>
<p>Note: if, for some n, f(n,d)=n for more than one value of d this value of n is counted again for every value of d for which f(n,d)=n.</p>
<hr>
<p><b>数字计数</b></p>
<p>从零开始的自然数在十进制中如下表示：<br>0 1 2 3 4 5 6 7 8 9 10 11 12….</p>
<p>考虑数字d=1。当我们写下数n之后，我们计算一下到目前为止数字1出现的次数，并记为f(n,1)。f(n,1)的前面一部分值如下所示：</p>
<table>
<thead>
<tr>
<th align="center">n</th>
<th align="center">f(n,1)</th>
</tr>
</thead>
<tbody><tr>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">2</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">7</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">8</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">9</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">10</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">11</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">12</td>
<td align="center">5</td>
</tr>
</tbody></table>
<p>注意到f(n,1)从不等于3。<br>等式f(n,1)=n的前两个解是n=0和n=1，下一个解是n=199981。</p>
<p>同样地我们记f(n,d)表示在写下n之后d出现的次数。<br>事实上，对于任意一个数字d ≠ 0，0都是方程f(n,d)=n的第一个解。</p>
<p>记s(d)是所有f(n,d)=n的解的和。<br>已知s(1)=22786974071。</p>
<p>对于1 ≤ d ≤ 9，求∑ s(d)。</p>
<p>注意：如果对于某些n，对于多于一个的d均满足f(n,d)=n，这个数n对于每个d都要计算一遍。</p>
<hr>

    </div>

    
    
    

      <footer class="post-footer">

        


        
    <div class="post-nav">
      <div class="post-nav-item">
    <a href="/155/" rel="prev" title="Problem 155">
      <i class="fa fa-chevron-left"></i> Problem 155
    </a></div>
      <div class="post-nav-item">
    <a href="/157/" rel="next" title="Problem 157">
      Problem 157 <i class="fa fa-chevron-right"></i>
    </a></div>
    </div>
      </footer>
    
  </article>
  
  
  



          </div>
          
    <div class="comments" id="gitalk-container"></div>

<script>
  window.addEventListener('tabs:register', () => {
    let { activeClass } = CONFIG.comments;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      let commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script>

        </div>
          
  
  <div class="toggle sidebar-toggle">
    <span class="toggle-line toggle-line-first"></span>
    <span class="toggle-line toggle-line-middle"></span>
    <span class="toggle-line toggle-line-last"></span>
  </div>

  <aside class="sidebar">
    <div class="sidebar-inner">

      <ul class="sidebar-nav motion-element">
        <li class="sidebar-nav-toc">
          文章目录
        </li>
        <li class="sidebar-nav-overview">
          站点概览
        </li>
      </ul>

      <!--noindex-->
      <div class="post-toc-wrap sidebar-panel">
          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="#Problem-156"><span class="nav-text">Problem 156</span></a></li></ol></div>
      </div>
      <!--/noindex-->

      <div class="site-overview-wrap sidebar-panel">
        <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
    <img class="site-author-image" itemprop="image" alt="sx349"
      src="/images/avatar.png">
  <p class="site-author-name" itemprop="name">Leonhard Euler(1707-1783)</p>
  <p class="site-description motion-element">Project Euler | 欧拉计划<br>中文翻译站</p>
</div>
  <div class="cc-license motion-element" itemprop="license">
    <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" class="cc-opacity" rel="noopener" target="_blank"><img src="/images/cc-by-nc-sa.svg" alt="Creative Commons"></a>
  </div>


  <div class="links-of-blogroll motion-element">
    <div class="links-of-blogroll-title"><i class="fa fa-link fa-fw"></i>
      友情链接
    </div>
    <ul class="links-of-blogroll-list">
        <li class="links-of-blogroll-title">
          <a href="https://projecteuler.net/" title="https:&#x2F;&#x2F;projecteuler.net" rel="noopener" target="_blank">Project Euler</a>
        </li>
        <li class="links-of-blogroll-title">
          <a href="http://jakwings.is-programmer.com/Project_Euler" title="http:&#x2F;&#x2F;jakwings.is-programmer.com&#x2F;Project_Euler" rel="noopener" target="_blank">饮水思源汉化</a>
        </li>
        <li class="links-of-blogroll-title">
          <a href="http://sx349.github.io/" title="http:&#x2F;&#x2F;sx349.github.io" rel="noopener" target="_blank">译者博客</a>
        </li>
    </ul>
  </div>

      </div>

    </div>
  </aside>
  <div id="sidebar-dimmer"></div>


      </div>
    </main>

    <footer class="footer">
      <div class="footer-inner">
        

        

<div class="copyright">
  
  &copy; 2001 – 
  <span itemprop="copyrightYear">2021</span>
  <span class="with-love">
    <i class="fas fa-calculator"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">Project Euler | 欧拉计划</span>
</div>
  <div class="powered-by">由 <a href="https://hexo.io/" class="theme-link" rel="noopener" target="_blank">Hexo</a> & <a href="https://mist.theme-next.org/" class="theme-link" rel="noopener" target="_blank">NexT.Mist</a> 强力驱动
  </div>

        








      </div>
    </footer>
  </div>

  
  <script src="/lib/anime.min.js"></script>
  <script src="/lib/velocity/velocity.min.js"></script>
  <script src="/lib/velocity/velocity.ui.min.js"></script>

<script src="/js/utils.js"></script>

<script src="/js/motion.js"></script>


<script src="/js/schemes/muse.js"></script>


<script src="/js/next-boot.js"></script>




  




  
<script src="/js/local-search.js"></script>













  

  
      

<script>
  if (typeof MathJax === 'undefined') {
    window.MathJax = {
      loader: {
        source: {
          '[tex]/amsCd': '[tex]/amscd',
          '[tex]/AMScd': '[tex]/amscd'
        }
      },
      tex: {
        inlineMath: {'[+]': [['$', '$']]},
        tags: 'ams'
      },
      options: {
        renderActions: {
          findScript: [10, doc => {
            document.querySelectorAll('script[type^="math/tex"]').forEach(node => {
              const display = !!node.type.match(/; *mode=display/);
              const math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
              const text = document.createTextNode('');
              node.parentNode.replaceChild(text, node);
              math.start = {node: text, delim: '', n: 0};
              math.end = {node: text, delim: '', n: 0};
              doc.math.push(math);
            });
          }, '', false],
          insertedScript: [200, () => {
            document.querySelectorAll('mjx-container').forEach(node => {
              let target = node.parentNode;
              if (target.nodeName.toLowerCase() === 'li') {
                target.parentNode.classList.add('has-jax');
              }
            });
          }, '', false]
        }
      }
    };
    (function () {
      var script = document.createElement('script');
      script.src = '//cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js';
      script.defer = true;
      document.head.appendChild(script);
    })();
  } else {
    MathJax.startup.document.state(0);
    MathJax.texReset();
    MathJax.typeset();
  }
</script>

    

  

<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/gitalk@1/dist/gitalk.min.css">

<script>
NexT.utils.loadComments(document.querySelector('#gitalk-container'), () => {
  NexT.utils.getScript('//cdn.jsdelivr.net/npm/gitalk@1/dist/gitalk.min.js', () => {
    var gitalk = new Gitalk({
	  title       : document.title.substr(0,29),
      clientID    : '4ba99a8839c5e7dde683',
      clientSecret: '3984f7cad409553d6afde8dbc2c08fc425e7bbdc',
      repo        : 'pe-cn-comments',
      owner       : 'pe-cn',
      admin       : ['sx349'],
      id          : '966b5faeae2c4cfd5bd153ffa406610c',
        language: '',
      distractionFreeMode: true
    });
    gitalk.render('gitalk-container');
  }, window.Gitalk);
});
</script>

</body>
</html>
